Auction method and apparatus

ABSTRACT

An automatic system for determining outcomes to an auction process represents the auction by a directed graph and uses a K best solutions algorithm to determine the K best solutions. The system uses a particular graphical representation. Constraints may be included directly into the graph.

FIELD OF INVENTION

The invention relates to apparatus for solving certain problems, including for example auction problems, and related methods.

BACKGROUND

A particular example of a winner determination problem is the winner determination problem for auctions. For example, consider a reverse auction in which a number of sellers provide bids for supplying quantities of a variety of goods and/or services. It would be desirable to have an automatic system capable of providing the best set of bids to accept.

A difficulty may arise in that often there will be some form of constraint on the process so that the winner determination problem is not simply a question of selecting the set of bids which generate the lowest cost. For example, there may be a desire to have at least two suppliers, to avoid over-reliance on a single supplier, but not too many suppliers, to avoid excessive costs in procurement and delivery. It may be very hard to mathematically represent such constraints. For example, it may be very difficult to represent the desire not to have “too many suppliers” since it may not be clear at the start of the auction how many suppliers represents “too many”.

RELATED APPLICATIONS Prior US Applications

U.S. application Ser. No. 11/232,518 filed 22 Sep. 2005 “Computing A Set of K-Best Solutions to an Auction Winner-Determination Problem” (HP Ref: 200502482) and

U.S. application Ser. No. 11/546,042 filed 11 Oct. 2006 “Constraint Satisfaction for Solutions to an Auction Winner-determination Problem” (HP Ref: 200600318).

BRIEF DESCRIPTION OF DRAWINGS

Embodiments will now be described, purely by way of example, with reference to the accompanying drawings, in which:

FIG. 1 is a schematic diagram of an auction system according to an embodiment;

FIG. 2 is a more detailed schematic diagram of the auction system of FIG. 1;

FIG. 3 is a schematic diagram of a graph representation of an auction system according to a comparative example;

FIG. 4 is a schematic diagram of a graph representation according to an embodiment;

FIG. 5 is a schematic diagram of an alternative graph representation according to another embodiment; and

FIG. 6 is a schematic diagram of an alternative graph representation according to another embodiment.

DETAILED DESCRIPTION

The invention will now be described with reference to a particular example. The particular example is to an auction system, and the example presented relates to a reverse auction, also known as a procurement auction, where a purchaser invites bids to supply a number of different goods or services, which will be referred to as items. In general, there will be S sellers and I such items.

The system operates schematically as shown in FIG. 1. A number of bids 120A, 120B, . . . 120N are submitted to the auction system 130, and a number of solutions (greater than one) are output as the k best solutions 180. Each bid indicates a quantity of an item and a cost of the item.

This is illustrated in more detail in FIG. 2, which illustrates the solutions module 210 within the auction system 130. This contains a formulation module 220 for representing the bids as a graph 140, stored on a computer-readable medium and a solutions determination module 230 for determining the solutions 180. A preprocessing module 231 is in the embodiment provided for preprocessing the graph 140 represented by the formulations module. The computer readable medium may be, for example, a memory or a readable disk.

The output may be a number k of solutions 180. The system may also generate subsets of these, such as the subset of feasible solutions 282 that comply with certain constraints. These will be discussed in more detail below.

The output may be displayed on a display 182, output to a computer readable medium 184, or both.

One of the solutions is then selected, and the corresponding set of bids in the path of that solution are accepted and the remaining bids rejected. The suppliers then supply the quantities of the accepted bids and are paid the amounts of the accepted bids.

The auction system 130 also allows the inputs of objective functions 240, for determining the goal of the solutions determination module (for example, lowest cost) and side constraints 250.

There is no obligation that each supplier supplies all of any given item. For example, one supplier may supply half the required number of a particular kind of item, and another the rest. Thus, each item may be subdivided into a number of shares. The number of shares will be denoted Q (for quantiles), and each item may therefore be divided into Q quantiles. For example, if Q=4, there are four quantiles, and each supplier may supply 0, 1, 2, 3 or 4 quantiles of each of the items, representing 0%, 25%, 50%, 75% and 100% of the total requirement for that item.

In this example, each seller provides a number of bids for each possible number of quantiles for each of the items. The bid from the s-th seller for supplying q_(s) quantiles of the i-th item will be denoted (q_(s)).

It will be appreciated that I, S and Q are integers. Since there are I items, i may take the values 1, 2, 3 . . . I, and likewise s may take the values 1, 2, 3 . . . S and q may take the values 0, 1, 2, 3 . . . Q, since it is possible to supply 0 items.

Given this significant number of bids, the problem is for the buyer to determine the best solution.

The approach adopted in the present invention is to represent this problem as a directed graph.

One version of such an approach has previously been considered within Hewlett Packard. In this approach the problem is broken down into sub-auctions for each of the items, a comparative example, illustrated in FIG. 3. There are two sellers (A and B) and the number of quantiles Q is 2, so that for each item there are two lots. Thus, for each item, the possibilities are that seller A sells both lots, represented as AA, that seller B sells both lots, represented as BB, and that each seller sells one lot, represented as AB.

The formulation module 220 constructs a graph as illustrated in FIG. 3, with a source node 310, a destination node 340 and a number of intermediate nodes 320, 330. In this case, there are two intermediate nodes, a first intermediate node 320 and a second intermediate node 330.

Each item is represented by a set of edges (the lines in FIG. 3) between two nodes, corresponding to the possible combinations to deliver each item. Thus, graph of FIG. 3 represents three items. The first item is represented by the edges between the source node 310 and first intermediate node 320, and each edge represents one of the three possible ways the two suppliers can supply the product. Similarly, the second item is represented by the edges between first and second intermediate nodes 320, 330, and the third item is represented by the edges between the second intermediate node 330 and the destination node 340.

The length of each edge is then given as the sum of the bids for that edge. For example, the length of the edge AB between source node 310 and the first intermediate node 320 is the sum of the bid of seller A for supplying one unit and the bid of seller B for supplying one unit.

The problem of finding the best combination of bids to accept then becomes the problem of finding the shortest path from the source node 310 to the destination node 340.

In this approach the number of edges between each pair of adjacent nodes, corresponding to each item, can be large. In particular, the number of edges for Q quantiles and S sellers is given by

$\begin{matrix} {N = \frac{\left( {Q + S - 1} \right)!}{{Q!}{\left( {S - 1} \right)!}}} & (1) \end{matrix}$

For significant numbers of sellers and quantiles, this number can be very large. Accordingly, the inventor has realised that an improved graph can significantly reduce the number of edges.

For each item, instead of a series of parallel edges between adjacent nodes as in FIG. 3, a more complex pattern of edges is adopted between source node s_(i) 400 and sink node t_(i) 410, via intermediate vertices 420 labelled (s, q) for s=1, 2, 3 . . . S and q=1, 2, 3 . . . Q. The source and sink nodes correspond to adjacent vertices 310, 320 in the comparative example of FIG. 3.

Consider first the specific example as illustrated in FIG. 4, which is for the case of a single item with three sellers and three quantiles (S=3, Q=3). The edges are directed left to right, and are labelled with their lengths.

Each vertex (s,q) represents a situation where the sellers from 1 up to s have supplied their bids and supplied a total of q quantiles. Thus, where each seller s supplies q_(s) units,

$q = {\sum\limits_{1}^{s}{q_{s}.}}$

Thus, the vertex (2,1) represents a situation where the first two sellers have in total supplied one unit.

The rows in FIG. 4 are labelled with the values of q and the columns with the seller.

The four edges from starting vertex s_(i) which may be represented as (0,0) represent the supply from the first seller of 0, 1, 2 or 3 quantiles respectively, and the lengths are the bids for supplying the respective number of quantiles. These edges terminate in the vertices (1,0), (1,1), (1,2) and (1,3) where the second number represents the total number of quantiles supplied.

From each of these four vertices, the edges represent the supply by the second seller of 0, 1, 2, or 3 quantiles respectively. However, it should be noted that if the first seller has already supplied 2 quantiles, the second seller can only supply 0 or 1 since in this example the total number of quantiles supplied is 3. Therefore, there are only two paths from the (s,q)=(1,2) vertex corresponding to the supply of zero and 1 quantile respectively by the second seller.

The set of paths leading to sink t_(i) (3,3) corresponds to the third seller supplying the remainder of the quantiles required at the corresponding price.

Thus, the different paths from source 400 to sink 410 represent the different ways that the three sellers can supply the three quantiles of the item in question concerned.

Of course, this approach may be used for other numbers of sellers and quantiles.

Expressed more generally and mathematically, the following algorithm may be used to generate the graph for S sellers and Q quantiles for each item.

The source node is s_(i) 400 and the sink node t_(i) 410, and intermediate vertices 420 are labelled (s, q) for s=1, 2, 3 . . . S and q=1, 2, 3 . . . Q.

For the first seller, s=1, for each quantile q₁ from 0 to q₁, an edge is added from the node s_(i) to vertex (1,q₁) labelled q₁ and with length B_(i1)(q₁).

For each of the sellers from s=2 to S−1, for quantiles q≦Q and q_(s)≦Q−q, connect the vertices (s−1, q) to (s, q+q_(s)) via an edge of label q_(s) and length B_(is) (q_(s)).

For the last seller s=S, connect each vertex (S−1,q) to the node t_(i) via an edge with label Q−q and length B_(iS) (Q−1).

At first sight, this graph may seem more complex than those of FIG. 3. However, the number of vertices is Q(S−1)+2 for each subgraph (i.e. for each item) and the number of edges is (S−2)((Q+1)(Q+2)/2)+2Q.

Since there are I items, in general, the total number of vertices V is given by

V=I(Q(S−1)+2)  (2)

and the total number of edges E is given by

E=((S−2)((Q+1)(Q+2)/2)+2Q)  (3)

The total number of vertices is O(ISQ)—that is to say of order (ISQ), and the total number of edges O(ISQ²).

By comparison with equation (1), it may be seen that this polynomial order is much better than the number in the FIG. 3 case, for large Q and S.

The particular graphical representation used can cope efficiently with significant numbers of sellers and of a significant number of quantiles, i.e. wherein the amount of each item can be subdivided into a large number of different pieces. Thus, the use of the representation as recited allows faster and more computationally efficient processing than alternative formulations in these cases.

The above description and FIG. 4 relate to a sub-graph for a single item 500 which is chained together with graphs for other items 510, 520 to provide the complete graph as illustrated in FIG. 5.

In an alternative approach the graph may be defined by a single set of rules for all items, rather than for each item separately.

In this approach, there is a vertex for each triple (i,s,q) where i represents an item, s the seller and q the number of quantiles already supplied of by sellers up to and including the s-th seller of the i-the item. For s=S q=Q. There is an extra source node s=(0,0,0) and sink t=(I+1,S,Q).

The graph is defined with edges as follows.

1. Each vertex (i,s,q) is connected to (i, s+1 q+q_(s)) for each s≦S−1 and q_(s)≦Q−q. These are the majority of edges and correspond to the assignment of q_(s) quantiles of item i to seller s. These edges are labelled q_(s) and have length B_(is) (q_(s)). 2. Each vertex in the form (i,S−1 q) is connected to (i,S,Q) for 0<i≦I. These edges correspond to allowing the last seller to supply all remaining quantiles of item i. These edges are labelled (Q−q) and have length B_(iS) (Q−q) 3. (i,S,Q) is connected to (i+1,1,q₁) or each 0≦i<I and q₁≦Q. These correspond to the assignment of q₁ items of the (i+1)—the item to the first seller, and thus represent the linking of the sub-graphs together. These edges have label q₁ and length B_((i+1)1)(q₁) 4. (i,S,Q) is connected to t with a single edge of length 0.

The graphs may be represented in the auction system 130 in any convenient way.

After the formulation module 220 has specified the graph, the solutions determination module 230 then determines a plurality (k) shortest paths.

The skilled person is aware of a number of “k—shortest path” (KSP) algorithms which may be used for finding the k shortest paths, and any of these may be used to identify the k best solutions 180 (FIG. 2)

The use of a k-shortest path algorithm instead of just finding the single shortest path has certain benefits. It has been appreciated that the best path according to the algorithm is not always the best path in the real world, since there may be additional constraints or considerations that are difficult to incorporate into the model. For this reason, the provision of a plurality (k) shortest paths allows the user to select from the suitable shortest paths to deal with other constraints, such as a general desire to use no more than a particular number of suppliers, or the desire to take at least some items from one or more preferred suppliers if this can be done without excessively increasing costs.

Such constraints can be very hard for users to capture mathematically. For example, the desire to use a preferred supplier if this can be done without excessively increasing costs requires knowledge in advance of what “excessively” means in this context, which may be hard to know in advance but which is easy for a procurement professional to identify presented with a list of options and their costs.

Referring back to FIG. 2, the solutions determination module can be arranged not just to provide the k best solutions 180 but also subsets of these solutions according to additional criteria determined by side constraints.

These can include features such as the requirement to use or not to use particular combinations of suppliers. The k best solutions 180 are searched to provide the feasible solutions that meet these side constraints, which will be a subset of the K best solutions.

The above embodiments do not code constraints into the model or into the solutions determined by the solutions determination model. The user simply selects suitable solutions from the k best solutions.

However, in other embodiments, which will now be described, some or all of the constraints are incorporated into the graphs so that they are automatically considered by the solutions determination module 230.

In general terms, the same approach is used as in FIG. 5 as described above with a single graph. Each vertex is additionally labelled with an additional index x that can take any of a set of values that will be denoted as the set X which is {*, 1, 2, . . . N} where * is a special element of the set, representing the initial state.

Thus, assuming that the set X has a certain number of possible values, each vertex (i, s q) for possible values of i, s and q in the graph of FIG. 5 may be replaced by that number of vertices, labelled (i,s,q,x) for each possible value of x.

The concept is to use the variable x to capture additional details to allow constraints to be modelled within the graph.

For example consider a case where only certain subsets of sellers may be used, for example no more than two sellers. For simplicity, there may be three sellers, 1, 2 and 3 of which no more than two may be selected to supply all items. Each vertex may be labelled with the set of sellers that have already supplied.

In this case the set X may be represented as {*, 1, 2, 3, 12, 13, 23}. * is the situation where no sellers have supplied any units, and each other element represents the sellers that have already supplied. Note that there is no element 123, since this is not permitted by the constraint that there be no more than two sellers. Further, note that the values of X are in this case non-numeric—they represent elements of a set. There is in general no requirement that x is a numeric value.

The source point 400 is labelled with x=*, and each edge as defined above connects (i,s,q,x) to (i′,s′,q′, x′) where the variable x′ keeps track of how many sellers have supplied. Thus, each edge from the source point (which represent the quantity of the first item sold by the first seller) is to x′={1} except for the case where q′=0, which mean that no items are sold from the first seller, which is to x′={*}.

More generally, an edge from x={1} which represents a sale by the second seller is to the vertex labelled x′={12} since “after” this additional sale both sellers 1 and 2 have supplied.

There are no edges defined from points labelled x={12} which involve supply by the third seller since this is not permitted. Such edges are omitted from the graph.

These omitted edges from the graph mean that there are “hanging” edges which do not form part of any path from source to sink in this embodiment. However, the KSP algorithms can readily cope with this, so alternative embodiments may not have this property.

FIG. 6 illustrates part of such a graph. It corresponds to the part starting from a value 1=0 and s=1 representing the supply of either zero or one unit by seller s=2. Each vertex of FIG. 4 is replaced by seven vertices in FIG. 6, corresponding to the seven possible different values of x: *, 1, 2, 3, 12, 13, 23.

The horizontal edges represent the supply of no units by seller 2. The value of x is accordingly unchanged.

The diagonal lines to q=1 represent the supply of one unit by seller 2. Thus, where x={1} at the start, indicating that only seller 1 has supplied any units, x′={12} at the end indicating that sellers 1 and 2 have each supplied at least one unit.

Note that there is no edge from x={13} since this would need to go to the element {123} which is not an element of X. Accordingly, this edge is omitted.

Note that the value x={*} is only relevant for the first sub-graph 500 (as shown in FIG. 5). The subsequent sub-graphs 510, 520 do not require this, since the paths through the first sub-graph 500 always represent supply by at least one of the sellers.

In more general terms, the graph is defined with edges as follows. For completeness, the whole set of rules is given.

The variable x can take values in the set X where X={x: x⊂σ′ and σ′εS} where S is a collection of allowed sets of sellers and a′ is an element of this collection S.

1. Each vertex (i,s,q,x) is connected to (i, s+1, q+q_(s), x′) for each s<S−1 and q_(s)≦Q−q for a particular value of x′ except where the edge is omitted as defined below. These are the majority of edges and correspond to the assignment of q_(s) quantiles of item i to seller s.

x′=x if the edge is labelled 0, x′=x∪{s+1} otherwise. If the edge is not labelled 0 and x′ as defined is not an element of X, the edge is omitted.

2. Each vertex in the form (i,S−1,q,x) is connected to (i,S,Q,x′) for 0<i≦I and a particular value of x unless the edge is omitted. These edges correspond to allowing the last seller to supply all remaining quantiles of item i. These edges are labelled (Q−q) and have length B_(iS) (Q−q)

Again, x′=x if the edge is labelled 0, x′=xÅ{S} otherwise. If the edge is not labelled 0 and x′ as defined is not an element of X, the edge is omitted.

3. (i,S,Q,x) is connected to (i+1,1,q₁,x′) or each 0≦i<I and q₁≦Q. These correspond to the assignment of q₁ items of the (i+1)-th item to the first seller, and thus represent the linking of the sub-graphs together. These edges have label q₁ and length B_((i+1)1)(q₁)

Again, x′=x if the edge is labelled 0, x′=x∪{1} otherwise. If the edge is not labelled 0 and x′ as defined is not an element of X, the edge is omitted.

4. (i,S,Q, x) is connected to t with a single edge of length 0 for those values of x in S. If x is not in S, the vertex is not connected to t.

Note that although in the simple example above for which X={*, 1, 2, 3, 12, 13, 23} all allowed values for x (except *) are in S, this is not necessarily the case. For example, if the constraint was that exactly two sellers need to be suppliers, S would be {12, 13, 23} whereas x can take on any of {*, 1, 2, 3, 12, 13, 23}.

Only paths corresponding to suitable sets for which the sellers comply with the requirement connect the source 400 to the sink 410.

A similar formulation can be used to include quantile thresholds in the graph, i.e. thresholds where there are constraints on the number of quantiles supplied by the sellers.

In this case X is a product of sets of the form {0, 1, 2 . . . T_(s)} for each seller s where T_(s) is a threshold, for example a maximum or minimum value for that seller. x can then be represented by (x₁, x₂, x_(s)) where x_(s) for each of s=1, 2 . . . S counts the number of quantiles so far assigned to seller s.

To implement a minimum, x_(s)′=min (T_(s),x_(s)+q_(s)) on a path labelled q_(s), i.e. a path corresponding to supplier s supplying q_(s) quantiles. x_(s) then counts up to the minimum requirement. Any seller already meeting this requirement has x_(s)=T_(s). Then, only those penultimate vertices (I,S,Q,x) are connected to the sink for which x_(s)=T_(s) for all suppliers. (Note that T_(s) can take the value 0 for some suppliers).

To implement a maximum, x_(s)′=x_(s)+q_(s). Where x′_(s) is greater than T_(s) the edge is simply omitted. In this case, all vertices (I,S,Q,x) are connected to the sink.

Thus, by using the above approaches involving the additional vertex label X, some constraints may be incorporated in the graph

In the examples above that deliver a plurality of solutions, i.e. the k-best solutions, it is possible for a human user of the system to determine a solution that meets constraints even if the constraints are not mathematically formulated.

It should be noted that the above embodiments are presented purely by way of example.

Please note that the above description is based on the same number Q for all items. However, in alternative embodiments the number of quantiles Q may vary for each item, in which case it may be represented as Q_(i) for each i from 1 to I.

Further, although the above description describes a reverse auction, exactly the same approach can be used for a normal auction for which the goal is to maximise revenue, by simply representing each bid by either by a negative number or by a constant less the cost of each bid. Thus, instead of the sellers in the embodiments described above, the participants may in fact be buyers.

The representation of the elements may vary. For example, the variable q may represent the percentage of the total supplied, not the number of quantiles.

Further, the bids may not be bids from external suppliers, but they may instead represent the cost of supply of a resource from an internal source. For example, instead of suppliers, each counterparty in the auction may be a computer server that supplies a particular amount of processing of a particular job and the bids represent the cost of that server supplying that amount of processing power. The auction outcome in this case represents the least expensive use of resources to achieve a particular processing goal.

For this reason, the term “counterparty” is used to refer to the participants in the auction—these counterparties may be buyers, sellers or even inanimate objects such as computer servers. 

1-8. (canceled)
 9. A method for evaluating auction bids in an auction with I items and S counterparties, where each item is subdivided into Q units, the method comprising: accepting a plurality of bids, each bid being a bid B_(is) (q_(s)) representing the cost of a quantity q_(s) of item i from counterparty s; generating, by executing a module stored on a non-transitory computer readable medium, a directed graph from a source vertex to a sink vertex representing the plurality of bids; carrying out a k-shortest path algorithm to identify the k shortest paths from source to sink, the bids making up each shortest path representing a corresponding solution to the auction problem and the k shortest paths representing the k best solutions to the auction problem, where k is an integer and k≧2; and outputting the k best solutions on a display or computer readable medium.
 10. A method according to claim 9 wherein the step of generating a directed graph comprises: generating the source vertex (0,0,0) and the sink vertex; generating a plurality of intermediate vertices (i,s,q) for each item i where i=1, 2, . . . I, for each counterparty s where s=1, 2, . . . S, and for each possible quantity q where q=0, 1, . . . Q, wherein when s=S and q=Q; and generating edges representing each bid, each edge being an ordered edge from the source vertex or one of the intermediate vertices towards the vertex (i, s, q) of length B_(is) (q_(s)) representing the cost of a quantity q_(s) of item i by counterparty s where q is the total quantity of units of item i supplied by counterparties from counterparty 1 up to and including counterparty s. 11-17. (canceled)
 18. A computer program product stored on a non-transitory computer readable medium for evaluating auction bids in an auction with 1 items and S counterparties, where each item is subdivided into Q units, the computer program product comprising: code for accepting a plurality of bids, each bid being a bid B_(is) (q_(s)) representing the cost of a quantity q_(s) of item i from counterparty s; code for generating a directed graph from a source vertex to a sink vertex representing the plurality of bids; and code for carrying out a k-shortest path algorithm to identify the k-shortest paths from source to sink representing the k best solutions to accepting bids to obtain each item, where k is an integer and k≦2.
 19. A computer program product according to claim 18 wherein the code for generating a directed graph comprises: code for generating the source vertex and the sink vertex; code for generating a plurality of intermediate vertices (i,s,q) for each item i where i=1, 2, . . . I, for each counterparty s where s=1, 2, . . . S, and for each possible quantity q where q=0, 1, . . . Q, wherein when s=S and q=Q; and code for generating edges representing each bid, each edge being an ordered edge from the source vertex or one of the intermediate vertices towards the vertex (i, s, q) of length B_(is) (q_(s)) representing the cost of a quantity q_(s) of item i by counterparty s where q is the total quantity of units of item i supplied by counterparties from counterparty 1 up to and including counterparty s.
 20. A computer program product according to claim 19, wherein code for generating edges comprises: code for providing an edge of length B_(is) (q_(s)) from each vertex (i, s, q) to (i, s+1 q+q_(s)) for each s<S−1 and q_(s)≦Q−q; code for providing an edge labeled (Q−q) of length B_(is) (Q−q) from each vertex (i,S−1, q) for 0<i≦I; and code for providing an edge labeled q₁ of length B(_(i+1))₁(q₁) from each vertex (i,S,Q) to (i+1, 1, q₁) for each 0≦i<I and q₁≦Q; the method further comprising identifying the source vertex as (0,0,0) and identifying the final vertex (I,S,Q) as the sink vertex or connecting the vertex (I, S, Q) to the sink vertex with a path of length
 0. 21. A computer program product according to claim 19, wherein each vertex is labeled with an additional variable x where x is an element of a set X, so that the vertices are labeled (i, s, q, x), wherein the variables x are used to represent a constraint on the solution.
 22. A computer program product according to claim 21, wherein every edge starting at (i, s, q, x) and labeled q₁ goes to (i, s+1, x′) where x′ is an element of X, x′ is a function of a rule representing the constraint which determines x′ from one or more of i, s, q, s, q′ and x; and any edges which do not meet the constraint are omitted.
 23. A computer program product according to claim 21 wherein the constraint is that the set of counterparties supplying non-zero quantities is an element of a set S of counterparties and wherein X is given by X={*; x: x⊂σ′ and σ═εS} where {*} is a special element, the method comprising: providing an edge of length B_(is) (q_(s)) connecting each vertex (i,s,q,x) to (i, s+1, q+q_(s), x′) for each s<S−1 and q_(s)≦Q−q for a particular value of x′ except where the edge is omitted as defined below, where x′=x if the edge is labeled 0, x′=∪{s+1} otherwise, and if the edge is not labeled 0 and x′ as defined is not an element of X, the edge is omitted; providing an edge labeled (Q−q) of length B_(iS) (Q−q) connecting each vertex (i,S−1,q,x) to (i,S,Q,x′) for 0<i≦I and a particular value of x unless the edge is omitted, where x′=x if the edge is labeled 0, x′=x∪{S} otherwise, and if the edge is not labeled 0 and x′ as defined is not an element of X, the edge is omitted; providing an edge labeled q₁ of length B_((i+1)1)(q₁) connecting each vertex (i,S,Q,x) to (i+1, 1,q₁,x′) for each 0≦i<I and q_(i)≦Q where x′=x if the edge is labeled 0, x′=x∪{1} otherwise, and if the edge is not labeled 0 and x′ as defined is not an element of X, the edge is omitted; and providing an edge connecting (i,S,Q, x) to the sink with a single edge of length 0 if and only if x is in S.
 24. A computer program product according to claim 21, wherein the constraint is a requirement that one or more counterparties s supply a minimum total quantity T_(s); X is a product of the sets {0, . . . , T_(s)) for each of the one or more counterparties s, the rule determining x′ is x′=min (T_(s), x+q₁); and of the vertices (I,S,Q,x), only those vertices for which x_(s)=T_(s) for each s are connected to the sink.
 25. A computer program product according to claim 21, wherein the constraint is a requirement that one or more counterparties s supply a maximum total quantity T_(s) and X is a product of the sets {0, . . . , T_(s)) for each of the one or more counterparties s.
 26. A method according to claim 10, wherein generating edges comprises: providing an edge of length B_(is) (q_(s)) from each vertex (i, s, q) to (i, s+1 q+q_(s)) for each s<S−1 and q_(s)≦Q−q; providing an edge labeled (Q−q) of length B_(is) (Q−q) from each vertex (i,S−1, q) for 0<i≦I; and providing an edge labeled q₁ of length B(_(i+1))₁(q₁) from each vertex (i,S,Q) to (i+1, 1, q₁) for each 0≦i<I and q₁≦Q; the method further comprising identifying the source vertex as (0,0,0) and identifying the final vertex (I,S,Q) as the sink vertex or connecting the vertex (I, S, Q) to the sink vertex with a path of length
 0. 27. A method according to claim 10, wherein each vertex is labeled with an additional variable x where x is an element of a set X, so that the vertices are labeled (i, s, q, x), wherein the variables x are used to represent a constraint on the solution.
 28. A method according to claim 27, wherein every edge starting at (i, s, q, x) and labeled q₁ goes to (i, s+1, q′, x′) where x′ is an element of X, x′ is a function of a rule representing the constraint which determines x′ from one or more of i, s, q, s, q′ and x; and any edges which do not meet the constraint are omitted.
 29. A method according to claim 27 wherein the constraint is that the set of counterparties supplying non-zero quantities is an element of a set S of counterparties and wherein X is given by X={*; x: x⊂σ′ and σ′εS} where {*} is a special element, the method comprising: providing an edge of length B_(is) (q_(s)) connecting each vertex (i,s,q,x) to (i, s+1, q+q_(s), x′) for each s<S−1 and q_(s)≦Q−q for a particular value of x′ except where the edge is omitted as defined below, where x′=x if the edge is labeled 0, x′=x∪{s+1} otherwise, and if the edge is not labeled 0 and x′ as defined is not an element of X, the edge is omitted; providing an edge labeled (Q−q) of length B_(iS) (Q−q) connecting each vertex (i,S−1,q,x) to (i,S,Q,x′) for 0<i≦I and a particular value of x unless the edge is omitted, where x′=x if the edge is labeled 0, x′=x∪{S} otherwise, and if the edge is not labeled 0 and x′ as defined is not an element of X, the edge is omitted; providing an edge labeled q₁ of length B_((i+1)1) (q₁) connecting each vertex (i,S,Q,x) to (i+1, 1,q₁,x′) for each 0≦i<I and q₁≦Q where x′=x if the edge is labeled 0, x′=x∪{1} otherwise, and if the edge is not labeled 0 and x′ as defined is not an element of X, the edge is omitted; and providing an edge connecting (i,S,Q, x) to the sink with a single edge of length 0 if and only if x is in S.
 30. A method according to claim 27, wherein the constraint is a requirement that one or more counterparties s supply a minimum total quantity T_(s); X is a product of the sets {0, . . . , T_(s)) for each of the one or more counterparties s, the rule determining x′ is x′=min (T_(s), x+q₁); and of the vertices (I,S,Q,x), only those vertices for which x_(s)=T_(s) for each s are connected to the sink.
 31. A method according to claim 27, wherein the constraint is a requirement that one or more counterparties s supply a maximum total quantity T_(s) and X is a product of the sets {0, . . . , T_(s)) for each of the one or more counterparties s.
 32. Auction apparatus for carrying out an auction having I items and S counterparties, each item being subdivided into Q units and accepting a plurality of bids, each bid being a bid B_(is) (q_(s)) representing the cost of a quantity q_(s) of item i from counterparty s, the apparatus comprising: a formulation module stored on a non-transitory computer readable medium and executed to generating a directed graph from a source vertex to a sink vertex representing the plurality of bids; and a solutions determination module to solve a k-shortest path algorithm to identify the k shortest paths from source to sink, the bids making up each shortest path representing a corresponding solution to the auction problem and the k shortest paths representing the k best solutions to the auction problem, where k is an integer and k≧2.
 33. An auction apparatus according to claim 32, wherein the formulation module generates the directed graph by generating a plurality of intermediate vertices (i,s,q) for each item i where i=1, 2, . . . I, for each counterparty s where s=1, 2, . . . S, and for each possible quantity q where q=0, 1, . . . Q, wherein when s=S and q=Q; and generating edges representing each bid, each edge being an ordered edge from the source vertex or one of the intermediate vertices towards the vertex (i, s, q) of length B_(is) (q_(s)) representing the cost of a quantity q_(s) of item i by counterparty s where q is the total quantity of units of item i supplied by counterparties from counterparty 1 up to and including counterparty s. 